A microscopic model for the self-inductance of an ideal solenoid

TL;DR

This paper provides a microscopic derivation of the self-inductance of an ideal solenoid by equating the magnetic energy $U$ to the total kinetic energy of drift electrons and computing the on-axis field via Biot–Savart and Ampère's law. The main result is $L = \frac{\mu_0 N^2 S}{\ell}$, derived from $U = \frac{\mu_0 N^2 S I^2}{2 \ell}$. The mass of the electron drops out, indicating geometry alone governs $L$ in the ideal limit, and the work offers a concrete physical picture of energy storage in inductors while reconciling historical misconceptions. The discussion also addresses limitations, including assumptions of infinitesimal wire thickness, infinitude, and ideal zero resistance, and reflects on pedagogical lessons via a postscript on Lagrangian formalism.

Abstract

We derive the formula for the self-inductance of an ideal solenoid by calculating the total kinetic energy associated with the drift velocity of the conduction electrons.